A California bill requiring the state’s pension funds to secure detailed fee disclosures from private-equity firms is awaiting the signature of Gov. Jerry Brown, in a sign of the mounting pressures the industry faces to become more transparent.

The bill, AB 2833, which requires public funds in the state to report how much they pay in annual fees, expenses and carried interest for each private-equity fund and hedge fund they back, received unanimous support in the state assembly. Gov. Brown has about a month to sign the bill...